MTH 4500: Introductory Financial Mathematics
# Midterm 2 (Practice 2)

**Problem 1**
**Problem 2**
**Problem 3**
**Problem 4**
**Problem 5**
**Problem 6**

Assume an one period binomial model in which the initial stock price is \( S=300 \) and in each period the stock price can go either up by a factor of \( u=\frac{7}{5} \) or down by a factor of \( d=\frac{2}{5} \). Assume that the simple interest rate over one time period is \( r=\frac{1}{5} \). Determine the price of the European call option with strike \( K=180 \).

Which of the following two options is more expensive: A European put option with strike \( 65 \) or a European put option with strike \( 70 \)? The options are derived from the same stock and have the same expiration.

Assume a two period binomial model in which the initial stock price is \( S=25 \) and in each period the stock price can go either up by a factor of \( u={\frac{12}{5}} \) or down by a factor of \( d={\frac{2}{5}} \). Assume that the simple interest rate over one time period is \( r=100\% \). Determine the price of the put option of strike \( K=45 \) with expiration \( T=2 \) that can be exercised either at time \( 1 \) or at time \( 2 \).

The intial price of the stock is \( S_0=17 \) and the price follows the binomial model in which in each period the price can go up by the factor \( u \) or down by the factor \( d \). The numbers \( u \), \( d \), and the interest rate \(r\) are not known to us. A company *Not Very Smart Bank Made Solely for The Purposes of This Problem (NVSBMSFTPOTP)* hopes to make money by trading the European call options on this stock with strikes \( 14 \), \( 26 \), and \( 35 \) and expiration \( T=20 \). It has published the following prices for which it is willing to buy and sell the options:
\[
\begin{array}{|c|c|c|}\hline \mbox{Strike} & \mbox{Bid} & \mbox{Ask}\newline
\hline
14 & 41 & 42\newline \hline
26 & 31 & 32\newline \hline
35 & 20 & 21\newline \hline\end{array}\]
Prove that there is an arbitrage opportunity and explain how this arbitrage can be obtained.

*Note.* The *ask* price is the price that you have to pay if you want to buy the particular option from NVSBMSFTPOTP. The *bid* price is the price that NVSBMSFTPOTP is willing to pay you if you want to be the writer of the option and NVSBMSFTPOTP the buyer.

Assume that the stock price follows a binomial model with \( n=30 \) steps. The intial price of the stock at time \( 0 \) is \( S_0=27 \) and in each step the price of the stock goes up by the factor \( u=\frac{4}{3} \) or down by the factor \( d=\frac{3}{4} \). The interest rate is assumed to be \( 0 \). Consider the derived security that pays USD \( 1 \) if the price of the stock at time \( n \) is exactly \( 48 \) and during the entire time interval \( [0,30] \) the price has always stayed below the level \( B=64 \). Determine the price of the described security.

Determine the number of sequences \(X_1\), \(X_2\), \(\dots\), \(X_{198}\) with terms in \(\{-1,1\}\) such that \(X_1+\cdots+ X_{198}=0\) and for each \(k\in\{1,2,\dots, 198\}\) the following holds: \begin{eqnarray*}\left|X_1+X_2+\cdots+X_k\right|< 50.\end{eqnarray*}